• Title/Summary/Keyword: Marcinkiewicz integral operators

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WEIGHTED ESTIMATES FOR ROUGH PARAMETRIC MARCINKIEWICZ INTEGRALS

  • Al-Qassem, Hussain Mohammed
    • Journal of the Korean Mathematical Society
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    • v.44 no.6
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    • pp.1255-1266
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    • 2007
  • We establish a weighted norm inequality for a class of rough parametric Marcinkiewicz integral operators $\mathcal{M}^{\rho}_{\Omega}$. As an application of this inequality, we obtain weighted $L^p$ inequalities for a class of parametric Marcinkiewicz integral operators $\mathcal{M}^{*,\rho}_{\Omega,\lambda}\;and\;\mathcal{M}^{\rho}_{\Omega,S}$ related to the Littlewood-Paley $g^*_{\lambda}-function$ and the area integral S, respectively.

ON CERTAIN ESTIMATES FOR ROUGH GENERALIZED PARAMETRIC MARCINKIEWICZ INTEGRALS

  • Daiqing, Zhang
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.47-73
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    • 2023
  • This paper is devoted to establishing certain Lp bounds for the generalized parametric Marcinkiewicz integral operators associated to surfaces generated by polynomial compound mappings with rough kernels given by h ∈ ∆γ(ℝ+) and Ω ∈ Wℱβ(Sn-1) for some γ, β ∈ (1, ∞]. As applications, the corresponding results for the generalized parametric Marcinkiewicz integral operators related to the Littlewood-Paley g*λ functions and area integrals are also presented.

SINGULAR AND MARCINKIEWICZ INTEGRAL OPERATORS ON PRODUCT DOMAINS

  • Badriya Al-Azri;Ahmad Al-Salman
    • Communications of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.401-430
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    • 2023
  • In this paper, we prove Lp estimates of a class of singular integral operators on product domains along surfaces defined by mappings that are more general than polynomials and convex functions. We assume that the kernels are in L(log L)2 (𝕊n-1 × 𝕊m-1). Furthermore, we prove Lp estimates of the related class of Marcinkiewicz integral operators. Our results extend as well as improve previously known results.

WEIGHTED ESTIMATES FOR CERTAIN ROUGH OPERATORS WITH APPLICATIONS TO VECTOR VALUED INEQUALITIES

  • Liu, Feng;Xue, Qingying
    • Journal of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.1035-1058
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    • 2021
  • Under certain rather weak size conditions assumed on the kernels, some weighted norm inequalities for singular integral operators, related maximal operators, maximal truncated singular integral operators and Marcinkiewicz integral operators in nonisotropic setting will be shown. These weighted norm inequalities will enable us to obtain some vector valued inequalities for the above operators.

A NOTE ON GENERALIZED PARAMETRIC MARCINKIEWICZ INTEGRALS

  • Liu, Feng
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1099-1115
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    • 2019
  • In the present paper, we establish certain $L^p$ bounds for the generalized parametric Marcinkiewicz integral operators associated to surfaces generated by polynomial compound mappings with rough kernels in Grafakos-Stefanov class ${\mathcal{F}}_{\beta}(S^{n-1})$. Our main results improve and generalize a result given by Al-Qassem, Cheng and Pan in 2012. As applications, the corresponding results for the generalized parametric Marcinkiewicz integral operators related to the Littlewood-Paley $g^*_{\lambda}$-functions and area integrals are also presented.

A Note on Marcinkiewicz Integral Operators on Product Domains

  • Badriya Al-Azri;Ahmad Al-Salman
    • Kyungpook Mathematical Journal
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    • v.63 no.4
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    • pp.577-591
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    • 2023
  • In this paper we establish the Lp boundedness of Marcinkiewicz integral operators on product domains with rough kernels satisfying a weak size condition. We assume that our kernels are supported on surfaces generated by curves more general than polynomials and convex functions. This generalizes and extends previous results.

Weighted Lp Boundedness for the Function of Marcinkiewicz

  • Al-Qassem, Hussain M.
    • Kyungpook Mathematical Journal
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    • v.46 no.1
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    • pp.31-48
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    • 2006
  • In this paper, we prove a weighted norm inequality for the Marcinkiewicz integral operator $\mathcal{M}_{{\Omega},h}$ when $h$ satisfies a mild regularity condition and ${\Omega}$ belongs to $L(log L)^{1l2}(S^{n-1})$, $n{\geq}2$. We also prove the weighted $L^p$ boundedness for a class of Marcinkiewicz integral operators $\mathcal{M}^*_{{\Omega},h,{\lambda}}$ and $\mathcal{M}_{{\Omega},h,S}$ related to the Littlewood-Paley $g^*_{\lambda}$-function and the area integral S, respectively.

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